TY - JOUR

T1 - Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytope

AU - Guessab, Allal

AU - Semisalov, Boris

PY - 2018/9/1

Y1 - 2018/9/1

N2 - In this paper, we consider the problem of approximating a definite integral of a given function f when, rather than its values at some points, a number of integrals of f over some hyperplane sections of simplices in a triangulation of a polytope P in (Formula presented.) are only available. We present several new families of “extended” integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of simplices, and which contain in a special case of our result multivariate analogues of the midpoint rule, the trapezoidal rule and the Simpson’s rule. Along with an efficient algorithm for their implementations, several illustrative numerical examples are provided comparing these cubature formulas among themselves. The paper also presents the best possible explicit constants for their approximation errors. We perform numerical tests which allow the comparison of the new cubature formulas. Finally, we will discuss a conjecture suggested by the numerical results.

AB - In this paper, we consider the problem of approximating a definite integral of a given function f when, rather than its values at some points, a number of integrals of f over some hyperplane sections of simplices in a triangulation of a polytope P in (Formula presented.) are only available. We present several new families of “extended” integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of simplices, and which contain in a special case of our result multivariate analogues of the midpoint rule, the trapezoidal rule and the Simpson’s rule. Along with an efficient algorithm for their implementations, several illustrative numerical examples are provided comparing these cubature formulas among themselves. The paper also presents the best possible explicit constants for their approximation errors. We perform numerical tests which allow the comparison of the new cubature formulas. Finally, we will discuss a conjecture suggested by the numerical results.

KW - Approximation

KW - Best constants

KW - Convexity

KW - Cubature

KW - Error estimates

KW - ELEMENT

KW - CONVEX

UR - http://www.scopus.com/inward/record.url?scp=85044368809&partnerID=8YFLogxK

U2 - 10.1007/s10543-018-0703-3

DO - 10.1007/s10543-018-0703-3

M3 - Article

AN - SCOPUS:85044368809

VL - 58

SP - 613

EP - 660

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 3

ER -